A Convex Cone is a Convex Set

Last updated Nov 1, 2022

# Statement

\mathbb{R}

and let

S \subset V

be a Convex Cone. Then

S

is a Convex Set.

# Proof

Suppose $\mathbf{u}, \mathbf{v} \in S$ and $a \in [0, 1]$ . If $a = 0$ or $a = 1$ we have that $a \mathbf{u} + (1-a) \mathbf{v} \in {u, v} \subset S$ . Otherwise $a \in (0,1)$ so $a > 0$ and $1-a > 0$ . Because $S$ is a Convex Cone

, we have that

a \mathbf{u} + (1-a) \mathbf{v} \in S

. Thus

S

is a Convex Set.

\blacksquare

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A Convex Cone is a Convex Set

# Statement

Vector Space

Convex Cone

Convex Set

# Proof

Convex Cone

Convex Set

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